Theorem nnssn0s 28713

Description: The positive surreal integers are a subset of the non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
nnssn0s	⊢ ℕs ⊆ ℕ0s

Proof of Theorem nnssn0s

Step	Hyp	Ref	Expression
1		df-nns 28707	. 2 ⊢ ℕs = (ℕ0s ∖ { 0s })
2		difss 4072	. 2 ⊢ (ℕ0s ∖ { 0s }) ⊆ ℕ0s
3	1, 2	eqsstri 3954	1 ⊢ ℕs ⊆ ℕ0s

This proof depends on syntax axioms:   ⊆ wss 3895   ∖ cdif 4045  {csn 4584   0_s c0s 28197  ℕ_0scn0s 28704  ℕ_scnns 28705

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-dif 4047  df-nns 28707

This theorem is used by:  nnssno  28714  nnn0s  28719  nnn0sd  28720  nnsgt0  28731